The invention relates to the determination of the polarisation directions for the fast and slow shear waves arising from shear wave splitting due to anisotropy.
There are generally two types of seismic waves used in seismology, namely so-called xe2x80x9cP-wavesxe2x80x9d or compressional waves in which the vibrations occur in the direction of propagation of the waves, and so-called xe2x80x9cS-wavesxe2x80x9d or shear waves in which the vibrations occur in a direction generally orthogonal to the direction of propagation of the waves.
A multicomponent geophone is a directional detector for seismic waves, which includes a vector measurement of the incoming wave. In the applications considered here, two of the geophone components are assumed to be aligned along arbitrarily chosen X and Y directions, generally parallel to the surface of the earth. It is also assumed that the incoming shear waves will generally arrive vertically (i.e. perpendicular to the surface of the earth) from below the geophones. As a result the particle motion within the wave is generally parallel to the surface of the earth, and is detected by the X and Y geophone components.
Furthermore, as will be explained below, the incoming shear waves may contain two components which are polarised (in terms of the direction of vibration) in two orthogonal directions, S1 (i.e. the fast shear S1 propagation direction) and S2 (i.e. the slow shear S2 propagation direction), and which are separated from each other by a time delay. This specification is concerned with the determination of these two directions and the travel time delay between the corresponding shear waves.
From ocean bottom or land multicomponent surveys using a P-wave source, it is possible to obtain measurements of the shear waves converted in the earth. These shear waves appear predominantly on the horizontal (X and Y) components of the multi component geophones. If the earth is isotropic with respect to the horizontal direction of wave motion, then a single shear arrival may be expected for each reflecting interface. If however, as is often the case, the earth behaves anisotropically with respect to the horizontal direction (for example, because a geological layer is polarised in a particular direction due to fracturing), then we can expect to record two separate shear wave arrivals from each reflecting interface, arriving at different times, having propagated with different velocities. These are usually termed the fast (S1) and the slow (S2) shear waves, corresponding to the first and the second arrivals, respectively. They are also characterised by having different polarization directions (i.e. directions of particle motion in the horizontal plane), which in most cases are considered to be approximately orthogonal to each other. It is assumed that this is the case here.
The shear wave splitting phenomenon is illustrated in FIG. 1, which depicts a shear wave arrival (S) that, at the start (A) of an anisotropic medium, splits into two separate shear waves (S1 and S2), having different polarisation directions and propagating separately with differing velocities until the end (B) of the medium. If from (B) onwards the medium is supposed to be isotropic, the two polarised waves will continue to travel separately but with the same velocity until they impinge upon the recording geophones. The amplitudes recorded on each of the horizontal components of the geophone depend upon the orientations of the S1 and S2 directions relative to the X and Y directions.
FIG. 1 gives a simple graphical description of the principle of shear wave birefringence, by only considering one anisotropic layer imbedded in an isotropic medium. However, in reality there are many reflecting boundaries that give rise to a number of shear arrivals polarised in the S1 and S2 directions. In addition, these S1 and S1 directions can change between the different anisotropic layers. In the applications considered here, the S1 and S2 polarisation directions are assumed to be constant with depth, over the analysing time window.
According to the invention, there is provided a method of determining the polarisation directions of the fast and slow shear waves arising from shear wave splitting due to anisotropy, said directions defining a natural coordinate system, the method comprising the steps of:
a) recording at least two components of each shear wave, in a recording coordinate system,
b) calculating the value of xcex8, being the angle of rotation between the natural coordinate system and the recording coordinate system, for which the Lp norm is minimised if p is less than 2, or maximised if p is greater than 2.
In one embodiment of the invention, said value of xcex8 is determined by calculating the value of the Lp norm over a range of incrementally varying values of xcex8, and selecting that value of xcex8 for which the Lp norm is appropriately minimised or maximised.
In a further embodiment of the invention, p is 4, and the value of xcex8 is determined analytically from an equation derived by differentiating the Lp norm with respect to xcex8.
The two recorded components of each shear wave are sampled, for example, at about 4 ms intervals.
Preferably, the fast and slow shear waves are recorded using two orthogonal geophones, arranged generally parallel to the surface of the earth.
The fast and slow shear waves may be produced from a single source.
Said source may be a P-wave source or it may be a single shear source.
Said shear wave components are conveniently horizontal components.
The invention also includes apparatus for carrying out the above method, and a computer readable medium carrying a computer program for carrying out the above data processing steps.